To determine the inverse of the function [tex]\( f(x) = \frac{x + 2}{7} \)[/tex], we will follow a step-by-step method to find the function that, when composed with [tex]\( f(x) \)[/tex], will return [tex]\( x \)[/tex]. The inverse function [tex]\( f^{-1}(x) \)[/tex] will essentially "undo" what [tex]\( f(x) \)[/tex] does to [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Start with the given function:
[tex]\[
f(x) = \frac{x + 2}{7}
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x + 2}{7}
\][/tex]
3. Swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex] to begin finding the inverse function:
[tex]\[
x = \frac{y + 2}{7}
\][/tex]
4. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
- Multiply both sides of the equation by 7 to clear the fraction:
[tex]\[
7x = y + 2
\][/tex]
- Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 7x - 2
\][/tex]
5. Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = 7x - 2
\][/tex]
6. Identify the function from the given choices:
- A. [tex]\( p(x) = 7x - 2 \)[/tex]
- B. [tex]\( q(x) = \frac{-x + 2}{7} \)[/tex]
- C. [tex]\( r(x) = \frac{7}{x + 2} \)[/tex]
- D. [tex]\( s(x) = 2x + 7 \)[/tex]
From the above steps, it is evident that the correct inverse function is:
A. [tex]\( p(x) = 7x - 2 \)[/tex]
Therefore, the correct answer is choice A.
